Gregory Saint Vincent, S.J.

Gregory St. Vincent, S.J. was born in Bruges, Belgium and died d 1667 in Gand. He studied mathematics under Christopher Clavius. Gregory was a brilliant mathematician and is looked upon as one of the founders of analytical geometry. He established a famous school of mathematics at Antwerp. Gregory deals with conics, surfaces and solids from a new point of view, employing infinitesimals in a way differing from Cavalieri. Gregory was probably the first to use the word exhaurire in a geometrical sense. From this word arose the name of "method of exhaustion," as applied to the method of Euclid and Archimedes. Gregory used a method of transformation of one conic into another, called per subtendas (by chords), which contains germs of analytic geometry. He created another special method which he called Ductus plani in planum and used in the study of solids. Unlike Archimedes, who kept on dividing distances only until a certain degree of smallness was reached, Gregory permitted the subdivisions to continue ad infinitum and obtained a geometric series that was infinite.

Gregory was the first to apply geometric series to the "Achilles" problem of Zeno (in which the tortoise always wins the race with the swift Achilles since he has an unbeatable head start) and to look upon the paradox as a question in the summation of an infinite series. Moreover, Gregory was the first to state the exact time and place of overtaking the tortoise. He spoke of the limit as an obstacle against further advance, similar to a rigid wall. Apparently, he was not troubled by the fact that in his theory the variable does not reach its limit. His exposition of the "Achilles" paradox was favorably received by Leibniz and by other geometers over a century later.

Gottfried Leibniz credits Gregory St. Vincent in the development of analytic geometry, Gregory St. Vincent. In his work Opus geometricum quadraturae circuli et sectionum coni (1647) Gregory St. Vincent's treatment of conics earns him the honor of being classed by Leibniz along with Fermat and Descartes as one of the founders of analytic geometry.
This Opus geometricum has four books: first, concerning circles, triangles and transformations; then geometric sums and the Zeno paradoxes with trisection of angles using infinite series; third the conic sections; and finally, his quadrature method, based on his "ductus plani in planum" method. The latter is a summation process using a method of indivisibles, in which St. Vincent introduces his "virtual parabolas."

The Greeks described a spiral using an angle and a radius vector, but it was St. Vincent and Cavalieri who simultaneously and independently introduced them as a separate coordinate system. In an article The Origin of polar coordinates, J. J. Coolidge refers to the priority dispute between Cavalieri and St. Vincent over their discovery. St. Vincent wrote about this new coordinate system in a letter to Grienberger in 1625 and published the process in 1647. On the other hand Cavalieri's publication appeared in 1635 and the corrected version in 1653.

Gregory St. Vincent was one of the pioneers of infinitesimal analysis. In his Opus geometricum (1649) he proposed a new ingenious method of approaching the problem of infinitesimals and he gave his propositions a direct rigorous demonstration instead of the reductio ad absurdum argument used previously. St. Vincent added an element not previously found in geometrical works because he connected the question with the philosophical discussions of continuum and the result of the infinite division.

St. Vincent summed infinitely thin rectangles to find a volume by a process he called "ductus plani in planum" (multiplication of a plane into a plane). It is practically the same fundamental principle as today's present method of finding a volume of a solid of integration.
St. Vincent applied his process to many such solids and found the volumes. He differed from Cavalieri's method since his laminas "exhaust" the body within which they are inscribed: they have some thickness. This was a new use of this term since it literally does "exhaust" the volume instead of finding the volume to some predetermined accuracy. While St. Vincent was not clear how to visualize the process, he certainly was nearer to the modern view than any of his predecessors. This led him to the concept of a limit of an infinite geometrical progression which ultimately supplies the rigorous basis for the calculus. St. Vincent gave the first explicit statement that an infinite series can be defined by a definite magnitude which we now call its limit.

St. Vincent was probably the first to use the word exhaurire in a geometrical sense. From this word arose the name "method of exhaustion" . . . he used a method of transformation of one conic to another, which contains germs of analytic geometry. St. Vincent permitted the subdivisions to continue ad infinitum and obtained a geometric series that was infinite. . . . He was first to apply geometric series to the "Achilles". . . moreover was first to state the exact time and place of overtaking the tortoise. (See Florian Cajori: A History of Mathematics. New York: Chelsea, 1985, p. 181-182.)

St. Vincent in Book II of Opus geometricum applies his infinite-series process to Zeno's Achilles paradox and is perhaps the first to do so successfully. Boyer's comment on the efforts of St. Vincent and his unfortunate assumption that he had squared the circle:

Although St. Vincent did not express himself with the rigor and clarity of the nineteenth century, his work is to be kept in mind as the first attempt explicitly to formulate in a positive sense-although still in geometrical terminology-the limit doctrine, which had been implicitly assumed by both Stevin and Valerio, as also, probably, by Archimedes in his method of exhaustion. He maintained that he had squared the circle . . . and received disdain from his contemporaries, his memory being rehabilitated by Huygens and Leibniz. On the other hand there can be no doubt that his work exerted a strong influence on many of the mathematicians of his time. (See Carl Boyer: History of The Calculus. New York: Columbia, 1939, p. 55-57.)